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Kinetic phenomena in layered conductors placed in a magnetic field
PHYSICS Physics
ELSBVIER
Reports
REPORTS
288 (1997) 3055324
Kinetic phenomena in layered conductors placed in a magnetic field Y.G. Peschansky B.I. Verkin Institute for Low Temperature Physics and Engineering, Ukrainian National Academy 47 Lenin Avenue, 310164 Kharkov, Ukraine
ofSciences,
Abstract
A theoretical study on galvanomagnetic, high-frequency and magnetoacoustic phenomena in layered conductors with quasi-2D electron energy spectrum of an arbitrary form is presented. It is shown that the specific character of the quasi-2D energy spectrum of charge carriers causes a series of peculiar effects. It is the orientation effect - the oscillation dependence of kinetic characteristics on the angle between a strong magnetic field and a normal to the layers, specific resonance effects and acoustic transparency of the layered conductors. In a wide range of magnetic fields Kapitza’s law (the linear growth of the magnetoresistance with increasing value of strong magnetic field) is valid. PACS:
72.15.Gd; 73.61.At; 74.70.Kn; 75.70.Cn
Keywords; Layered conductor; electromagnetic field.
Magnetoresistance;
Acoustic
transparency;
Orientation
effect; Penetration
depth of the
1. Introduction A considerable success in the construction of the theory of condensed media is associated with the utilization of the concept of quasiparticles - elementary excitations above the basic state. The electron theory of metals developed by I.M. Lifshitz with the most general assumptions concerning charge carriers’ dispersion law [l] has stimulated an expensive complex of experimental investigations of thermodynamics and kinetic characteristics of metals placed in an external magnetic field. In a strong magnetic field these characteristics, being calculated under the assumption that the dependence of the conduction electron energy E on the momentump is known a priori, turn out to be very sensitive to the form of the electron energy spectrum. As a result the inverse problem formulated by I.M. Lifshitz can be solved by experimental study of the magnetic susceptibility and kinetic effects in metals in a strong magnetic field when the charge carriers rotation frequency C2 exceeds significantly their collision frequency l/r. The problem is to restore the Fermi surface E(P) = EF - the principal characteristic of the electron energy 0370-1573/97/$32.00 0 1997 PII SO370-1573(97)00030-6
Elsevier Science B.V. All rights reserved
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spectrum. In the momentum space the surface on which the energy is equal to the Fermi energy + turns out to have a complicated form and is open for almost all metals. Galvanomagnetic effects being the most sensitive to the Fermi surface topology have become the basis for constructing the reliable spectroscopic method for studying the topology structure of the electron energy spectrum [2,3]. The well-grounded concept concerning conduction electrons in metals is quite applicable to describe electron properties of a wider category of conductors in which the life time of excitations carrying a charge is large. This idea of I.M. Lifshitz proves to be fruitful for investigations of low-dimensional organic conductors. This is because “disorder” is not a common characteristic for organic conductors and a number of impurities and crystal defects is negligible. In such conductors, at low temperature, free path time of the charge carriers which realize the Fermi branch of the energy spectrum is sufficiently large and the condition fir 4 1 can be realized easily. A considerable part of organic conductors represent layered structures with a sharp anisotropy of electrical conductivity. The conductivity along a normal to the layers n is extremely small and much less than the conductivity along the layers. The latter, as a rule, is of the metal type and because of that the organic conductors are often named “organic metals” or synthetic metals. There are grounds to make use of the quasiparticle concept developed by I.M. Lifshitz for studying transport phenomena in such conductors. The sharp anisotropy of the conductivity of layered conductors is apparently connected with the sharp anisotropy of the velocity of charge carriers on the Fermi surface, i.e. their energy s(P) = C s,(~.+,) cos(anpJh) is weakly dependent on the momentum projection pZ =pn. The coefficients of cosines are assumed to decrease essentially with n, so that Ai = +4o < AI);
A,+1 4 A,;
(2)
where A, is the maximum value of the function E,(P,,P,,) on the Fermi surface, a is the separation between the layers and h the Planck constant. Specific character of the quasi-two-dimensional electron energy spectrum causes a series of the peculiar effects which are absent in the case of ordinary metals. It is the orientation effect - the oscillation dependence of kinetic characteristics on the angle between a strong magnetic field and a normal to the layers, specific resonance effects and acoustic transparency of the layered conductors. We shall consider theoretically galvanomagnetic phenomena and propagation of electromagnetic and acoustic waves through the layered conductors at an arbitrary form of a quasi-twodimensional charge carriers energy spectrum.
2. Galvanomagnetic
phenomena in layered conductors
Experimental observation in a magnetic field about lo-20 tesla of Shubnikov-de Haas oscillations of the magnetoresistance of such layered conductors as the tetrathiafulvalen salts, halogens of tetraseleiumtetracen [4411], indicates that the charge carriers’ mean free path I = m in them is sufficiently large, at least it is about lop3 cm. This gives grounds to suppose that the range of
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strong magnetic fields where galvanomagnetic characteristics are sensitive to the Fermi surface topology, can be reached easily. Below we analyse the dependence of magnetoresistance of the layered conductors on the magnitude of the magnetic field H = (0, Hcos 9, H sin 9) and its orientation with respect to the layers in the quasiclassic approximation, when the corrections to the magnetoresistance connected with the quantization of the charge carriers energy in a magnetic field are small. In order to find out the connection between the current density euif(p) 2d3 p(2xh)- 3 = oij(H)Ej;
ji =
(3)
s
and for the electric field E it is necessary to solve the kinetic equation for the charge carriers distribution function f(P) W + e(vxH)lc)V@
= JJL~{f).
(4)
However, at a small electric field the deviation of the distribution function f(P) =&(E) 11/(P)%&)/aE from the equilibrium Fermi functionfo(e) is small and the kinetic equation (4) can be linearized in a small perturbation of the conduction electrons system. In this approximation the collision integral WColrepresents a linear integral operator acting on $(p), which is the function to be found. At low temperatures when conduction electrons are scattered mainly by impurities and crystal defects we may take the collision integral to be the operator of multiplication by the collision frequency l/z of the unequilibrium correction to the Fermi functionfo(E). So the solution of the kinetic equation is the proper function of the integral operator of collisions. In this approximation the kinetic equation takes the simple enough form h+bjC%,+ $17 = eEv;
(5)
and its solution I,!/= eEi$i = eEi
t dt’ Vi(t’) exp (t’ - t)/z s -‘X
allows to determine the components
(6)
of the electrical conductivity
tensor
where e is the electron charge, tH is the time of electron motion in a magnetic field with the period T = 27c/!CJ according to the equations dpx/dt = (eH/c) { uycos 8 - v, sin 9};
ap,/at = -
(eH/c) u,cos 9;
ap,/at = (eH/c)v, sin 9; (8)
c is the velocity of light, angle brackets denote the and multiplication by 2eH/c (2~/2)~, subscript “H” of t discussion. It is easily seen that the velocity of motion of conduction the momentum projection pH = pZcos 9 + pp sin 9 onto the
integration over the Fermi surface is omitted here and in the following electrons at 11< 1 depends weakly on H-direction, and closed electron orbits
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in the momentum space are almost indistinguishable at different pH. It follows from this that the expansion in a power series with respect to the parameter of quasi-two-dimensionality y1of the conductivity tensor components begins from the quadratic terms if even one of the subscripts at Oij coincides with z [12]. It is easy to see that there are no terms proportional to y in the components giz = -
f n=l
x
7 2e3 Han/c(2n;h)3 h dp, dw(cp,l i i0
exp{(t’ - t)/r} sin {anp,/hcos
8 -
t
s ~ cx
dt'@,
tan 9/h).
anp,(t',p,)
PHI
(9)
The functions E, with n 2 1 is of the order of either ye or higher powers of y. In the linear approximation there is no need to account for the weak p,-dependence of the electron velocity, momentum projection pY and the functions E,. When integrating with respect to pH of the last factor, each of the terms in the sum in the formula (9) vanishes. The asymptotic behaviour of the components Ozj is analogous because in the expression for the velocity Uj(t,PH) = Vj(t) + Atij(t,PH) the corrections which depend on pH are proportional to q, and the velocity projection U, averaged over &, at fixed t is equal to zero in the linear approximation in q. An experimental observation of only two frequencies of the Shubnikov-de Haas oscillations of the magnetoresistance of the layered conductors [4-71 which corresponds to the minimum and maximum Fermi surface cross-sections, indicates that one group of charge carriers in such conductors prevails. The small difference between the frequencies is the result of the small corrugation of the Fermi surface. At least there is no reason to assume the compensation of the electrons and holes volumes in the presence of the several cavities of the Fermi surface to be permissible. When considering the galvanomagnetic phenomena, it is sufficient to take the Fermi surface in the form of a weakly corrugated cylinder with the open direction along the axis pz. The resistance of a conductor along the layers at an arbitrary orientation of a magnetic field is of the same order of magnitude as the resistance of a noncompensated metal, i.e. it differs unessentially from the resistance in the absence of a magnetic field which equals l/o0 q2. The difference from metals shows up in the larger amplitude of the Shubnikov-de Haas oscillations of the magnetoresistance of the layered conductor because the number of charge carriers which form the oscillatory effect increases proportionally to q- 112with decreasing v].At q -%hOI+ almost all of the charge carriers on the Fermi surface contribute to the oscillatory amplitude because all Fermi surface cross-sections of the plane PH are indistinguishable at yl = 0. On the contrary, the resistance of the layered conductor along the “hard” direction, pz,, i.e. along a normal to the layers, is sensitive to the magnetic field orientation. The asymptotic expression for pz,, as a rule, equals to l/o0 and only for some orientations of a magnetic field, at which crZZ< o,&, is determined by a greater number of the conductivity tensor components and increases sharply. Just in this the orientation effect lies. The asymptotic expression for cZZ(q,y) 0
--co
s T
dtl exph/d
dtdt)dt
0
+ tl) , i
(10)
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/ Physics Reports 288 (1997) 305-324
at yeQ 1 and y = l/L?z < 1 can be analysed easily at the most general assumptions about the function c, which satisfy the condition (2). Restricting our attention to the terms proportional to y2 we integrate over pH easily. As a result we have t
dt’ E,(t) c,(t’) (~n/h)~exp(t - t')/z -72
x [e” H cos S/iia~(27rh)~]cos ((an/h) (p,(t) - p,(t’)) tan $1,
(11)
where all functions under the integral sign depend on t and t’ only. For y < 1 the component electrical conductivity tensor ozz takes the following form fJ
zz
=
ae3zTHcos$/4n2h4cC
n
n21,f + y200{y2~1(9)
+ y2cp2($)},
of the
(12)
where
s I-
Z,(S) =
T-l
dt e,(t) cos {anpy(t) tan 9/h).
(13)
0
The functions qi are of the order of unity and account for them is necessary at the values 9 = 9, at which I1 (9) vanishes. This is the case when the asymptotic expression for the magnetoresistance depends essentially on the velocity of decreasing with n of the functions a,. For example, if I, is proportional to 11”the resistance along a normal to the layers at 9 = SC, instead of saturation in a strong magnetic field, increases proportionally to H2 in the range where y < y < 1 and the saturation of the resistance takes place in a more strong magnetic field at y 5 ye. For tan 9 $ 1 the expression under the integral sign in the formula (13) oscillates rapidly and I,($) can be derived easily by means of the stationary phase method. If there are only two points of the stationary phase where 0, vanishes in an electron orbit the asymptotic expression for Z,(9) takes the form 1,(S)
= 2eJO) (2xhc/aneHv!JO)
sin $1 ‘I2 cos(unD,
tan $/2h - n/4),
(14)
where D, is the Fermi surface diameter along the axis pY, and Pmin = p,(O), v:(O) > 0. Zeros of the function I1 (9) as it follows from the formula (14) repeat periodically with the period A(tan 9) = 2nh/aD,. Additional possibilities for studying the anisotropy of the Fermi surface diameters by the galvanomagnetic measurements are related to the presence of electron orbits strongly along the axis pz which intersect a large number of cells in the momentum space. The motion along such orbits grows as 9 approaches 7c/2 and can exceed z at any value of a field. Therefore, it should be kept in mind that the formulas given above are valid at not values of tan 9. The condition T < z can be satisfied only for electrons on the orbits which the points of self-crossing, for the period diverges as logarithm when approaching tan 9 2 l/q the self-crossing electron orbits appear. Analyse their role for 8 = x/2. The
(15) means of stretched period of magnetic too great have not them. At period of
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electron motion takes the form
along the open cross-sections
s
of the Fermi surface by the plane PH = PY = const.
2nh;a
T(P,) =
dp,
c/eHv,.
(16)
0
The velocity of the charge motion For example av,(t)pt
=
along the open orbit in this case at r] < 1 depends weakly on t.
(d2~~ap:)ap,~at+ (a2E~ap,dp,)ap,~at
= ww) (~,a~,lap, - v,av,iai4
- Y.
(17)
The period of motion on the orbits distant from the self-crossing orbits, is proportional to l/v,(O) within the linear approximation in q (t is supposed to be counted out from the cross section of the Fermi surface pz = 0). However, while approaching the self-crossing orbit pY = pC the velocity along the axis x decreases and account for the small corrections in the parameter v becomes necessary. In the point of self-crossing of the electron orbit pC = (O,p,,O) the velocity v,(O) vanishes and an electron stays near this point for a considerable time. Small value of U, on the electron orbit on which py is closed to pC, corresponds to the weak p,-dependence of E. In this case, we can make use of the expansion of energy in a power series in px while calculating the period of electron motion. Omitting the high-order modes in the formula (1) we obtain a(P) = so(0, P,) + PSR
+ a1(0, P,) cos @PZlh).
Making use of the relation (18) it is easy to calculate orbits close to the self-crossing trajectories
(18) the period
s
of electron
motion
along the
?I
T(p,)
= Y]~“~SZ~’
da(c2 + sin’ a)- 1’2,
(19)
0
where Q. is the frequency to the layers, and
of rotation
of an electron
in a magnetic
t2 = {a - &o(O,P,) - ~l(O>P,))/2~l(O>P,).
field applied parallel to a normal (20)
When approaching the self-crossing orbit, < becomes however small and the integral in the expression (19) diverges proportionally to ln(l/[). In distinction from ordinary metals in which the period of motion of charge carriers is larger or comparable with the free path time only in the small range (of the order of exp{ - sZoz>) of the Fermi surface cross-sections near the self-crossing section, in the quasi-two-dimensional conductor the condition T 2 z is satisfied in a more wide range of electron orbits where t is less or of the order of unity. in ozz. At y”’ 5 y. = 1/C20z the small part of the charge carriers makes the major contribution The velocity along the normal of these electrons can be written in the form v, = - ~~(0, p,) a/h sin C2i t, where fil = aeHv,(O)/hc h/aml.
= C20v,/vF and vr is the characteristic
(21)
Fermi velocity being of the order of
V.G. Peschansky i Physics Reports 288 (1997) 305-324
As it is easy to calculate,
the conductivity
along a normal
311
to the layers has the form
The contribution to crZZdue to electrons on the closed orbits of the Fermi surface is essentially smaller than (22). So the small range of the open orbits near the self-crossing orbit makes the major contribution in the electrical conductivity of a specimen, and the resistance of a conductor along a normal to the layers in the given above range of a magnetic field increases proportionally to H. Further increase of a magnetic field leads to narrowing of the range of electron orbits where the period of motion of an electron is larger than its free path time. As a result the contribution to crZZfrom the electrons near the self-crossing orbits is proportional to y; 2 as well as the contribution of electrons for which T Q r, and at y. < y112 the resistance pZZ increases proportionally to H 2. Thus in layered conductors with the quasi-two-dimensional electron energy spectrum there is a wide range of magnetic fields where the resistance along a normal to the layers increases proportionally to the magnetic field.
3. High-frequency
phenomena in layered conductors
Propagation of electromagnetic waves in layered conductors with a quasi-two-dimensional electron energy spectrum depends essentially on the polarization of the incident wave. A linear polarized wave with the electric field E,(r) along the normal to the layers decays at large distances than a wave with the electric field along the layers. This permits utilizing layered conductors as filters letting the wave through only with a certain polarization. Under the normal skin-effect conditions, when the mean free path 1 of the charge carriers is less than skin depth, the penetration depth 6,, of the electric field E,(r) is l/q times the penetration depth 6, of the electric field along the layers E,(r). When the mean free path of a conduction electron is larger than 61, the relation between 61 and 6,, is 6, =
y2136 II’
(23)
In a magnetic field the wave penetration depth and relation between 6, and 6,, depends essentially on the value and orientations of the strong magnetic field (Gr 9 1). Let us consider propagation of electromagnetic waves along the layers within the x > 0 halfspace in a magnetic field H = (0, H sin 8, H cos 9) parallel to the sample surface x, = 0. The electric field E(x) is to be found using the Maxwell equation curl curlE
- 02E/c2 = 47ciwjlc’;
div E = 4rcp’,
where p’ is the uncompensated charge density. Connection between current density and electric field can be found from the Boltzman equation for the charge carriers distribution function
(24) kinetic
(25)
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We confine our consideration to a linear approximation system. In this case, the kinetic equation is of the form
a+/at, + v,a$/ax + tj(i/~ -
in the weak perturbation
of the electron
iw) = evE(x),
(26)
and current density j is a linear integral operator acting on the function E(x). When solving the Maxwell equation, the wave can be reasonably treated as monochromatic with frequency w. The linearized kinetic equation (26) should be supplemented with the boundary condition allowing for the scattering of charge carriers by the sample surface
where the sample surface specularity W(p,p + ) through the expression &_)
= 1-
d3pW(p,p+)
parameter
s(J_)
is related
to the scattering
(1 - @ [G.(P)]),
indicatrix
(28)
s
where O(c) is the step-function, the electron momentump_ andp. (incident and scattered by the boundary at point x, = 0) are interrelated by the specular reflection condition, which conserves the energy and the momentum projection on the yz plane. The integral term in the boundary condition (27) ensures no current through the sample surface. This functional of the scattering indicatrix should be found with the aid of a rather intricate integral equation [13]. However, in the range of high frequencies co the solution of the kinetic equation depends weakly on this functional and without account for it has the following form at x < (x(tff) - x(0)): $(tfi,hf,
4 =
si
Cl{ dt4t,~FM(4t~~H)
+ 4(hAU
- LH))
- x(i~d)ev(v(t
- 4(khdexp{Ww
-
U)lY
?T-L
X
J i
dtev(t,pH)E(x(t,pH) - x(ApH))exp{v(t-
where v = - ice + l/z, and A is the root of the equation x(t,PH)
- XhpH)
nearest
to
tH
+ 2%- T)},
(29)
tH:
(30)
= x.
For the electrons that do not collide with the sample surface, i.e. for x > {x(tH,pH) - X,in) one should put i, = - CC. Following Reuter and Sondheimer [14], we continue evenly the electric field onto the region of negative x and apply the Fourier transformation to Eq. (24). As a result, the relationship between the Fourier transforms of the electric field and the current density
s cc
E(k) = 2
dx E(x) cos kx,
0
j(x) = 2
Ccdxj(x) cos kx, s0
(31)
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313
take the following form {k2 - Co2/c2}E,(k)
- 4Soj,(k)/c2
= - 2EL(O), CI= (y, z).
(32)
Apart from Eq. (32) the relation between j(k) and E(k) can be found by solving the kinetic equation (26), viz., ji(k) = oij(k)Ej(k)
+
s
where
dk’ Qij(k, k’) Ej(k’),
(33)
t ss s T
aij(k) = 2e3 H/c(~Tc~)~ dp,
dt’ uj(r’,PH)
dtui(t,PH)
0
x
exp{v(t’ - t)}
cos
-cc
k{x(t’,PH)
- x(t,PH)} G (e2ViRUj).
(34)
The integral operator kernel Qij depends essentially on the state of the sample surface. In a strong enough magnetic field parallel to the surface of a conductor, when the trajectory diameter 2~ of electrons along the x axis is much smaller than the skin-layer depth and the charge carriers mean free path 1,the relation between current density and inhomogeneous electric field can be to a good accuracy treated as local and independent on the sample surface condition since the current due to electrons colliding with the sample surface is negligible. In this case the skin depth can be accurately enough determined from the roots of the dispersion equation det{&a - 5G&)/c2}
= 0,
a,B =
(35)
(Y,z),
where t = 4rcio/(k2c2 - 02) and &&) = o,p(k) - o,,(k) o,#lMk).
(36)
With CM5 1 the decay length d1 pertaining to the electric field E,(x) coincides in order of magnitude with & = (oz)-~‘~ c/o0 for an arbitrary relation between 6,, and 1 and for any orientation of the magnetic field within the yz plane, since r?,,(k), to within numerical factors of order unity, coincides with go which is the electrical conductivity along the y axis in homogeneous electric field at H = 0. Here o. is the frequency of plasma oscillations of the electron gas. The electric field decay length 61, along the normal to layers depends essentially on the angle 9 between the vectors IZand H. This stems from the fact that at certain values of 9 there is an abrupt decrease of the expression for ozz(k), when kr < 1 and G?r9 1. The asymptotic expression for o,,(k) in a very strong magnetic field takes the following form: o,,(k) = 2
n21i(9) ue3 H cos Ip/cl/(2~h)~
II=0
+
nor2
{Y~~‘PIW
+
(W2
4n2(9)
+
~~v3(9)),
(37)
where y = v/L?; 1,(g) is given by the expression (13), the functions vi(S) are of the order of unity.
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At Svalues essentially differing from lp, (we recall that Ii (SC) = 0) there exists a universal relation between 6,, and dL, if q > (co/c~)~‘~, namely, to within a numerical factor of order unity
When the Fermi surface corrugation depth 61, = (1 + co2r2/c2)-1’2
is very small, namely,
6; o/c
q < min {(w/ao)1iZ,60/r},
the skin
(39)
increases with increasing magnetic field to reach the limiting value 6; w/c. The penetration depth for the electric field E,(x) grows substantially for 9 = gC, i.e., in the angular dependence of the impedance there is a series of narrow spikes at 9 = SC,, which in sufficiently pure conductors (1~~ > 6,) diminish with increasing magnetic field. On the other hand, they grow proportional l&,/ry, if ly2 < & and ly -=zr < do/y. At not too high frequencies, when the displacement current is small compared to conduction current across the layers, skin depth ~5)~ can be represented by the following interpolation formula (40) In the case of essentially depth 6,1 has the form
low conductivity
611= { 1 + (r/l~)~ + (rco/cq)“}
across the layers, when o > 00y2(q2 + r2/12), the skin
- lj2 &/y2,
(41)
and in a strong magnetic field, when r < (12y2 + 6i/q 2 ) I” , the electric field decay length across the layers is again equal to &/q2. The narrow spikes of the high frequency electric field E, with tan 9 $ 1 periodically appear as a function of the angle 9 with the period (15). Under the extremely anomalous skin effect conditions, when the skin depth is the least parameter of dimension length in the problem, the impedance and transparency of thin plates are sensitive to the state of the surface of a sample. If the surface is rough and it reflects charge carriers diffusely a universal relationship (23) between d1 and 6,, applies in a wide range of magnetic fields. When a specularity reflecting surface is parallel to a magnetic field, the relationship (23) is then valid only in the range of a weak magnetic field (r > l), whereas in a strong magnetic field (r < 1) we have 61 = 6,,qY
(42)
The relationship (42) is valid also in the case of near-specular reflection, when the effective width of charge carriers scattering indicatrix w < r312/18 f,‘2. If w + r3i2/ld:12 and 6, 4 r 4 1, then Eq. (23) applies. In the intermediate case when r312/16i’2 $ w 6 r3’2/1d:12, only 6, depends on w: 6,, = r 1’3(&/Y)2’3,
d1 = (w~)~/~S:/~ rp’j5.
(43)
It therefore follows that measurements of the impedance as a function of the magnetic field can be used to determine the state of the sample and the degree of anisotropy of the Fermi surface.
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When the skin layer depth 6 is considerably less than the diameter of an electron orbit in a magnetic field, an electromagnetic field penetrates in a conductor also in the form of narrow spikes, separated by distances that are multiples of the extremal diameter of the electron orbit (15). The weak curvature of the Fermi surface along the pZ axis favours participation of a large number of charge carriers in these spikes. If ye< 6/r, then almost all the conduction electrons with the energy E = FF participate in the penetration of an electromagnetic field deeper into a sample and intensities of such field spikes are of the same order of magnitude, with the exception of the last spike closest to the X, = 0 boundary of a thin plate (d 4 1), opposite to the skin layer, because on approach toward this boundary to a distance A (where 6 6 A 4 r), a small factor (A/r)“6(&,r)2i3 { 1 + wl/r}l13 appears due to the screening of the field in a spike by the current of electrons slipping along the boundary [16,17]. Investigation of the surface impedance in a magnetic field will allow determination in finer detail of the electron energy spectrum in layered conductors and the state of the sample surface.
4. Acoustic transparency of layered conductors When acoustic waves propagate in the layered conductor placed in a magnetic field, the anomalous transparency should be expected as well. Being very sensitive to the form of electron energy spectrum, the magnetoacoustic effects [l&21] have been used successfully for restoration of the Fermi surface, and in the low-dimensional conductors they are worthy of the special examination. In conducting crystals apart from the sound waves attenuation related to the interaction between thermic phonons and coherent phonons with the frequency co. there are many mechanisms of electron absorption of acoustic waves. The most essential of them is the so-called deformation mechanism connected with the charge carriers energy renormalization in the deformed crystal: KE=
AijUij.
(44)
In a magnetic field the induction mechanism - Joule losses - connected with electromagnetic fields generated by sound waves compete with the deformation mechanism. These fields should be derived from the Maxwell equations (24), and connection of the current density with the deformation tensor Uij and electric field E = E + (ti x H)/c + mii/e
(45)
can be found with the aid of the solution of the Boltzman kinetic equation. The field g is determined in the concominant system of axes which moves with the velocity ti. The last term in the expression (45) is connected with the Stuart-Toulmen effect. Together with the Maxwell equations and Boltzman kinetic equation it is necessary to consider the dynamic equation of the elasticity theory for the ionic displacement U. The equation contains the force which represents the influence on the crystal lattice of the electron system excited by the acoustic wave. In a weakly deformed crystal the complete set of equations for this problem is suggested by Silin [22] for isotropic metals and generalized by Kontorovich [23] to the case of an
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arbitrary dispersion law of charge carriers. A complete set of nonlinear equations for this problem, valid for any wave intensity, was derived by Andreev and Pushkarov [24]. In the approximation that is linear in the powers of the deformation tensor the change of the charge carrier dispersion law (44) can be described with the aid of the deformation potential 3Lijwhose components depend on the quasimomentump only and coincide in order of magnitude with the characteristic energy of the electron system, viz., the Fermi energy CF. Confining ourselves only to the linear approximation in a weak perturbation of conduction electrons under the action of deformation of the crystal in kinetic equation, we obtain for $ the following expression: $ = I?
{Aij(p)
Uij
+
eEv>,
(46)
where I? is the resolvent of the Eq. (27) and /lij(p)
=
&j(p)
-
/(l>.
The condition for existence of nontrivial solution of the complete set of the linearized equations for the problem represents the dispersion relation between the wavevector k and the frequency tr). The imaginary part of the wavevector determines the decrements of the acoustic and sound waves and the real part accounts for the renormalization of their velocities related to the interaction of the waves with conduction electrons. However, the sound attenuation rate can be also determined with the aid of the dissipation function Q which is proportional to the alteration with time of the entropy of a conductor [25]. Taking into account the electron absorption of acoustic waves only, we have for the dissipation function (47) and for the sound damping decrement
r = (I II/12/Pi2 ST),
(48)
where p is the density of the crystal, s the sound velocity, the collision integral is taken in the z-approximation. Using the Fourier representation for the Maxwell equations we obtain the connection between the electric field and the displacement of ions: (49) where /Lij= 6ij - kikj/k2, 6ij is the Kronecker symbol I?, and Ep are the projections of the electric field vector on the plane orthogonal to the sound wavevector. The electric field along k can be found from the condition of absence of the electric current along the sound wavevector, sinj is the anti-symmetrical tensor (s123 = 1). The components of the tensor oij(k) and aij(k) which connect the Fourier transformations of the current density, electric field and ionic displacement ji(k) = oij(k) Ej(k) + aij(k) kouj(k),
V.G. Peschansky 1 Physics Reports 288 (1997) 305-324
317
are of the following form: CJij(k)
=
(e2ViRUj);
aij(k)
=
(euiRAjn)
ujk”/k.
(51)
The components a,j and GaBconnect j, with Uj and Ep taking into account the absence of the current along the wavevector. Let us consider an acoustic wave propagating in the layers-plane in the x-direction orthogonal to a magnetic field H = (0, sin 9, cos 9). In this case when the parameter yeis small the solution of Eq. (49) takes the form: E,(k) = (1 - ~6yy(k)}-1 [icou,Hcos9/ck2
+ lkoiiyj(k)uj],
i?‘,(k) = - icou, H sin S/ck2 + ma’u,/e,
(52)
(53)
where ii,j(k) = aij(k) - a,j(k) ai,(k)/o,,(k). Expression (53) is valid when y2 < /3 = (oc/coO s)“/coz. In the opposite case, in order that E, be determined it is necessary to account for the dependence on q terms in the expansion of j, in a power series of the small parameter q. Nevertheless, at however small y there is no need to account for EZ because 0, E, is proportional to q, and it is sufficient to take into consideration the electric field E, only when calculating the asymptotic expression for the dissipation function at q = 0. In ordinary metals Joule losses are essential in the range of strong enough magnetic fields when the radius of curvature of the electron trajectory is much less than not only its mean free path but also the sound wavelength, i.e., kr 4 1. If the charge carriers trajectories are bent not too strongly, so that 1 < kr < kl,
(54)
the absorption of the sound wave energy in a metal is determined mainly by the deformation mechanism. In low-dimensional conductors the role of electromagnetic fields generated by a sound wave turns out to be essential in a more wide range of magnetic fields, including a magnetic field which satisfies the condition (54). In this case acoustoelectronic coefficients aij and Oij oscillate with an inverse value of a magnetic field and their asymptotic behaviour at 1 4 kr < l/q and P < I takes the following form: 6Jk)
= (G/kD)(l - sin kD);
a”yj(k) = - i(G/lj,/evkD)co~ kD,
(55)
where D = cD,/eH cos 9 and D, is the diameter of the Fermi surface along the p,-axis, 21and /ij, are the electron velocity and value of ijx(p) at the reference point along the p,-axis, G = 4vD,e2z/ac(2nh)2. At 1 < kD 4 Z/ythe asymptotic expression for the dissipation function of the sound wave with the
longitudinal
Q=
polarization
takes the form:
eHz cos S{g?(l + sin kD) + g$(l - sin kD) - 29, g2 cos kD) n2h2 akvc{l + ~~~,,,~‘}
where g1 = Aj,UjUk and g2 = VCOU,~Hcos $1~.
(56)
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/Physics
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It is easily seen that the asymptotic behaviour of the acoustoelecronic coefficients gYv and a”yjalters essentially at kD = 2n(n + i), which leads to the sharp increase of the dissipation function. However, out of the resonance, i.e. when cos kD differs from zero essentially and there is no need to take into account small corrections in the formulas (55) for the acoustoelectrical coefficients, the denominator in the expression (56) for the dissipation function increases with a magnetic field proportionally to H2 and the sound damping decrement has the form r E (co/v)r/l.
(57)
In this case the decrement decreases with increasing of the magnetic field magnitude and the mean free path of charge carriers. The attenuation length for a longitudinal wave is very large, when kD is close to 2~c(n- b), because g1 E g2 kD is much larger than g2. As a result, in addition to resonance we can expect the emergence of acoustic transparency when the path between two points of a stationary phase changes by half of the wavelength. As a result, between resonant values of the sound decrement which repeat with the period A (l/H) = 2ne cos $/kcD,
(58)
the anomalous acoustic transparency should be observed with the same period. In sufficiently pure conductors at low temperatures the charge carriers mean free path can be so large that the condition kln $ 1 is satisfied, there is the range of a magnetic field where l/q < kr + kl, and the oscillations of the acoustoelectric coefficients are formed by not all of charge carriers on the Fermi surface but by the small part of them. These are the charge carriers near the Fermi surface cross-section which has the extreme diameter Dixt. In this case the magnetoacoustic resonance at kl_H is absent, and the sound decrement T(H) = (1 + (kry)-‘12 sin(kD, - r-c/4)}ozjr
(59)
is of the same order of magnitude as in an ordinary metal. Unimportant numerical factors in the formulas (57) and (59) are omitted and Do = cD,““‘/eH cos 9. The magnetoacoustic resonance is possible at k_LH only if kry < 1. In the expressions for g,,v it is necessary to take into account the small corrections in the parameters r/l and l/kr near the resonant values of a magnetic field. It is not difficult to obtain the following formula for T(H) in the resonance:
At 1 < kr2 the absorption of the sound wave energy by conduction electrons increases proportionally to H with increasing magnetic field. The absorption of transverse waves in a layered conductor depends essentially on the concrete form of the nondiagonal components of the deformation potential tensor /lij(p). In the cases when in the range of the effective interaction between charge carriers and the sound wave where kv = CO,& and /i,, are much less than A,,, the order of magnitude of the sound damping decrement changes essentially. The periodic alternation of the transparency and the resonant absorption of the shear waves polarized in the layer-plane is of the same character as in the case of longitudinal waves.
V.G. Peschansb
1 Physics Reports 288 (1997) 305-324
319
Acoustic waves polarized along a normal to the layers attenuate at considerably larger distances if the component /1,, is small and vanishes at q + 0. In this case the solution of the Maxwell equations takes the form E
=
Y
kwii,,
+
mw2cYy,/e
1 - Gyy
E, =
4%
a",,ko(
+ mco”/e
1 - &?,,
u.
It is easy to see that the components of the matrix aij as well as oij do not contain the linear in y terms if at least one of the subscripts i or j coincides with z. When calculating the dissipation function, there is no need to retain the electric field E, which is proportional to y2 because ev,E, contains the term proportional to the first power in r. When 1 < kr <
Q=
co3m2eH cos 9 acsv(27ch)2
iAi ammo
/1,, +
hk(l - 55:,,)
cos (aD, tan 9/h) 2 (1 + sin kD).
When calculating Q we restrict ourselves to the first and second terms in the expression (1) for the charge carriers dispersion law. The acoustic transparency takes place when kD equals strictly to 27c(l - 4) and it is necessary to retain the small corrections in “/and l/kr. If sin kD differs essentially from ( - 1) the attenuation of sound waves increases with increasing H as so as in ordinary metals. The attenuation is determined mainly by the first term in the brackets in the formula (62) except for some exotic models of the deformation potential for which /1,, is much less than q&F.The amplitude of the oscillations of r with the period (58) which are caused by the periodic dependence of f7ZZ= y2go/kl { 1 + sin kD cos (uD, tan S/Jr);
(63)
is less than the smooth part of r in v/s times if /1,, r q&F.Besides, the decrement oscillates with 9 with the period (15), tan 9 need not be much more than unity. In the magnetic field conditioned by (54) the angular oscillations take place in the whole range of angles between the magnetic field vector and a normal to the layers. If an electron drifts along the sound wavevector (for instance, the sound wave propagates along the y-axis) the sound decrement reduces in (klq)2 times for r/l < krq < 1. The solution of the kinetic equation in this case taken the form t+T I) =
{exp(vT + &CT) - l}-’
t
s
dt’g(t’)exp(ik(r(t’)
- r(t)},
(64)
where g(t) = /Iji(t)kiuj + ev(t)E”. At 1 Q kly < l/r in the expansion in the powers of VT and kS_T = jt dt b(t) of the factor which is in front of the integral, the terms proportional to k are the most essential. Finally, in the case when the charge carriers drift along k with the velocity Vy= V,tan 9 E qv tan 9 should be replaced VT by kry tan 9 in the expression for the dissipation function. If kry tan 9 9 1, i.e. for the free path time an electron is capable to drift along the sound wavevector at distance which exceeds significantly the sound wavelength, the magnetoacoustic resonance predicted and studied theoretically by Kaner, Privorotsky and the authors of this paper [20] takes place. The resonance occurs at 6T and in contrast to an ordinary metal the amplitude of the resonant oscillations is determined by the parameter krq rather than kr.
KG. Peschansky / Physics Reports 288 (1997) 305-324
320
The formulas given above are valid when cos 9 $ cD,/eHl. If 9 is close to rc/2, i.e. cos 9 is so small that an electron have no time to make a total rotation along the orbit in a magnetic field, then the components of the tensors (T,,,,and crZZare closed to their values in the absence of a magnetic field. This results from the fact that in the quasi-two-dimensional conductor only the projection H, effects on the charge carriers dynamics, and at the q 4 1 the component H, manifests itself only in small corrections in the parameter ye. At 9 = 42 the dependence of the sound damping decrement on the magnetic field magnitude contains only the terms which vanish when r~+ 0, and the magnetoacoustic effects are pronounced in the case of the shear wave with the ionic displacement along a normal to the layers only. In the range of sufficiently strong magnetic field (kr Q 1) the attenuation rate for the wave with such polarization depends essentially on the magnitude of a magnetic field and its orientation with respect to the layers, and in the S-dependence of r sharp peaks and valleys appear. At tan 9 9 1 they are repeated periodically with the period (15). The concrete form of the 9dependence of r is in a great measure analogous to the angular dependence of the electromagnetic impedance for kr < 1. When the acoustic waves propagated along the normal to layers the Maxwell equations have the form {1 - 56:,,(k)} E, - 56,,(k) B, = rfi,j(k) Uj - (u, H, - u, HY) io/c - mo2u,.e; -
t&,,(k) E, + { 1 - 55,,(k)} Ey = [a”yj(k) Uj + u,H, io/c - mo2u,/e.
It is easy to see that the electric field and the components of the matrix gGlado not vanish when q - 0 and, consequently, the induction mechanism of the sound waves attenuation is more essential. The drift of conduction electrons along the z-axis does not take place only for the magnetic field orientation in the layers-plane. If 9 is not equal to 42, at krq Q 1 the displacement of charge carriers along the wavevector for the period of motion in a magnetic field is much less than the sound wavelength and the acoustoelectronic coefficients are of the same order of magnitude as the analogous values for the case of weak spatial dispersion being reduced in klq times if kly p 1. The essential dependence of r on H occurs only in the range of magnetic fields when krq % 1. At tan 9 < leH/cD, an ordinary magnetoacoustic resonance takes place. The resonance is connected with the charge carriers drift along the sound wavevector. At 9 = 42 electrons drift in the plane xy only, i.e., the direction orthogonal to the wavevector. In this case the sound attenuation rate oscillates with l/H which is analogous to the Pippard oscillations [21] in metals T(H) = ycoz/r { 1 - (krr])- “’ /?sin(kcdD,,/eH
Measurements
+ 7c/4)}.
(66)
of the period of these oscillations
d(l/H) = 4ne/kcdD,,,
(67)
enable the corrugation of the Fermi surface to be evaluated. Here d is the difference between the maximum and minimum diameters along the axis pXat pY= 0. The condition kry % 1 is very strict and can be satisfied in the range of a magnetic field where r <<1 only for q 2 &. Therefore, there are no grounds to expect that the clear dependence of r on the magnitude and orientation of a magnetic field can be observed in the layered conductors nowadays.
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Charged elementary excitations in conductors form a Fermi liquid, and their energy spectrum is determined by the distribution function for quasiparticles. As a result, the response of the electron system in solids to an external perturbation depends to a considerable extent on the correlation function describing the electron-electron interaction [27,28]. Usually, the inclusion of the Fermiliquid interaction of charge carriers leads to a renormalization of kinetic coefficients calculated under the assumption that conduction electrons form a Fermi gas. In some cases, however, the Fermi-liquid interaction approach leads to specific effects such as spin waves in nonmagnetic metals [29] and “softening” of metals in a strong magnetic field [30]. We shall show that the inclusion of the Fermi-liquid interaction of charge carriers depends significantly on the shape of the resonance curve and the value of acoustic transparency, but it does not violate the position of peaks of the acoustic wave absorption as a function of the value and orientation of a magnetic field [31,32]. The energy of elementary excitations carrying a charge has the form E = E(P) +
iLij(p)
Uij
+
Y(p,
Y,
t),
(68)
where the last term in this formula takes into account the correlation electronelectron interaction
effects associated with
Y b, r, t) = 2(27ch)- 3 @(p,p’) 6f(p’, Y,t) d3p’,
(69)
s
where 6f=f(p,r, t) -fo{~(P)} is th e nonequilibrium correction to the Fermi distribution function fo{s(P)} for charge carriers in the undeformed conductor. The Landau correlation function @(p,p’) can be expanded into the complete set of orthonormal functions 4”(p): (70) Confining ourselves only to the linear approximation in a weak perturbation of conduction the following equation for the nonequilibrium correction electrons we obtain - $(p,u)exp{ - iwt} 8fo(s)/& to the equilibrium distribution functionfo(s) for charge carriers in the concomitant reference frame moving with the velocity of ions: vrl/ + varl//ar +
a$/at, =
evE - i0 {
Aij
{p}
Uij
+
Y
(p,
r)}.
(71)
Using the Fourier representation y(P, 4 = g dk K(k) 6, (P) cxp {ikr) n=Os
(72)
we obtain the following system of algebraic equations for the Fourier transforms YJk): Y,,(k) (1 + @i ‘> + ia =
-
ikjui(k)(&(p)
(
hh-4~
Aij(p)>
f n=l
+
($n(P)
4A.p)
Y,,,(k) >
aCevg:(k)
+
kjaA,j(p)
u,(k)]),
(73)
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where {ik[v(t’) - r(t)] + v(t’ - t)}.
‘dt’g(t’)exp
Rg =
(74)
s
The value of wr is smaller than unity even in pure conductors at low temperature in a wide acoustic frequency range, and the integral term in formula (73) can be taken into account in the perturbation theory. In the asymptotic approximation in the small parameter OX, the Fourier transform of the kinetic equation solution $(k) assumes the form $(k) =
R{t?Ej(k)Uj
+
kiCO/iji
#j(k)}
@,(l +@“)-i [(4nReEjUj)
- ioR” z
- ikiuj(+,Aji)
+ o($nAji)kiUj]
+n
(75)
n=l
and the acoustoelectric coefficients gij(k) =
(e2viR”jV)- ioe2 $ @,(l + @,))I (ViR4n) (4nR”vj), ?l=l
Uij(k) = (eviR”Aj,)
km/k -
f
@,(I + @n)-’ (evi&J
(76)
((4nAjm)kmlk
n=l
+
im(4,i?Aj,
> k,/k}.
(77)
Using the expressions (76) and (77) we can easily determine the damping decrement for the acoustic wave. For example, we consider the influence of Fermi-liquid interaction on the attenuation rate of sound in the case when k = (k,O,O) and displacement of ions is in the plane xy. For brevity of computations only, we assume that 4r (-p) = 4l (p) and c$~(-p) = - ~$~(p),while the remaining & with YE> 2 are equal to zero. Taking into account that a( -p) = E(P), at ye< krq -@ 1 we obtain for rrY,,the following expression: liquid gYY
=
Ggas yy
(1
-
(78)
L),
where L
=
ioQ
A$:(1
+sinkD)+&4:(1
-sinkD)},
(79)
1
and c#J~,cj2 and v are the values of the functions 4i(t) and the velocity modulus when ku = CO. Naturally, the acoustic transparency of layered conductors with a quasi-two-dimensional electron energy spectrum and the sound attenuation rate depend on the intensity of the Fermi-liquid interaction of charge carriers, but the period of oscillations of r with l/H and the positions of sharp peaks in the angular dependence of r(9) remain unchanged when Fermi-liquid effects are taken into account.
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5. Conclusions The concept of quasiparticles developed by I.M. Lifshitz to describe electronic properties of metals have been used for investigations of kinetic phenomena in layered conductors with a quasi-two-dimensional electron energy spectrum. It is shown that the quasi-two-dimensional nature of the spectrum leads to the specific effects in a magnetic field. The magnetoresistance of layered conductors in a strong magnetic field directed in the plane of the layers can increase proportionally to a magnetic field value. The electromagnetic impedance and the sound attenuation rate depend essentially on the polarization of the incident wave and the angle between a magnetic field and a normal to the layers. Propagation of electromagnetic and acoustic waves in these conductors involves virtually all charge carriers in the transfer of acoustic pulses and electromagnetic field spikes to the bulk of the conductor. The orbits of Fermi electrons in a magnetic field are virtually indistinguishable, which allows to include a large number of conduction electrons in the formation of peculiar oscillatory and resonant effects which are absent in the case of ordinary metals. Investigation of these effects open up the possibilities of studying in detail the dissipative processes in the electron system of layered conductors and the charge carriers’ energy spectrum.
References [l] [2] [3] [4] [S] [6] [7] [S] [9] [lo] [l l] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
I.M. Lifshits, Selected Works. Electronic Theory of Metals. Polymers and Biopolymers, Nauka, Moscow, 1994. I.M. Lifshits, V.G. Peschansky, Zh. Eksp. Teor. Phys. 35 (1958) 1251 [Sov. Phys. JETP 8 (1958) 8751. I.M. Lifshits, V.G. Peschansky, Zh. Eksp. Teor. Phys. 38 (1960) 188 [Sov. Phys. JETP 11 (1960) 1371. M.V. Kartsovnic, P.A. Kononovich, V.N. Laukhin, IF. Shchegolev, Pis’ma Zh. Eksp. Teor. Fiz. 48 (1988) 498. I.D. Parker, D.D. Pigram, R.H. Friend, M. Kurmo, P. Day, Synth. Met. 27 (1988) A387. K. Oshima, T. Mori, H. Inokuchi, H. Urayama, H. Yamochi, C. Saito, Phys. Rev. B 38 (1988) 938. N. Toyota, T. Sasaki, K. Murata, Y. Honda, M. Tokumoto, H. Bando, N. Kinoshita, H. Anzai, T. Ishiguro, Y. Muto, J. Phys. Sot. Japan, 57 (1988) 2616. W. Kang, G. Montambaux, J. R. Cooper, D. Jerome, P. Batail, C. Lenoir, Phys. Rev. Lett. 62 (1989) 2559. I.F. Shchegolev, P.A. Kononovich, M.V. Kartsovnic, V.N. Laukhin, S.I. Pesotskii, B. Hilti, C.W. Mayer, Synth. Met. 39 (1990) 357. M.V. Kartsovnic, V.N. Laurhin, S.I. Pesotskii, I.F. Shchegolev, V.M. Yakovenko, J. de Phys. 1 France, 2 (1992) 89. S. Uji, H. Aoki, M. Tokumoto, A. Ugawa, K. Yakushi, Physica B 194 (1994) 1307. V.G. Peschansky, J.A. Roldan Lopez, Toji Gnado Yao, J. de Phys. I (France) 1 (1991) 1469. V.G. Peschansky, Sov. Sci. Rev. A. Phys. 16 (1992) 1-112. G.E.H. Reuter, E.H. Sondheimer, Proc. Roy. Sot. London 195 (1948) 336. M.Ya. Azbel’, Zh. Eksp. Teor. Fiz. 39 (1960) 400 [Sov. Phys. JETP 12 (1961) 2831. M.A. Lur’e, V.G. Peschansky, K. Yasemides, J. Low Temp. 56 (1984) 277. V.G. Peschansky, S.N. Savel’eva, H. Kheir Bek, Fiz. Tverd. Tela, (St.-Petersburg) 34 (1992) 1640 [Solid State Phys. 34(5) (1992) 8641. A.B. Pippard, Phil. Mag. 2 (1957) 1147. V.L. Gurevich, Zh. Eksp. Teor. Fiz. 37 (1959) 71. E.A. Kaner, V.G. Peschansky, I.A. Privorotsky, Zh. Eksp. Teor. Fiz. 40 (1961) 214. V.L. Gurevich, V.G. Skobov, Yu.D. Firsov, Zh. Eksp. Teor. Fiz. 40 (1961) 786.
324 [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
V.G. Peschansky / Physics Reports 288 (1997) 305-324 V.P. Silin, Zh. Eksp. Teor. Fiz. 38 (1960) 977. [Sov. Phys. JETP 11 (1960) 7751. V.M. Kontorovich, Zh. Eksp. Teor. Fir. 45 (1963) 1633. [Sov. Phys. JETP 18 (1963) 13333 A.F. Andreev, D.I. Pushkarov, Zh. Eksp. Teor. Fiz. 89 (1985) 1883. A.I. Akhiezer, Zh. Eksp. Teor. Fiz. 8 (1938) 1338. V.M. Gorhfel’d, O.V. Kirichenko, V.G. Peschansky, Zh. Eksp. Teor. Fiz. 108 (1995) 2147 [Sov. Phys. JETP 81 (1995) 11711. L.D. Landau, Zh. Eksp. Teor. Fiz. 30 (1956) 1058 [Sov. Phys. JETP 3 (1956) 9201. V.P. Silin, Zh. Eksp. Teor. Phys. 33 (1957) 495 [Sov. Phys. JETP 6 (1957) 3871. V.P. Silin, Usp. Fiz. Nauk, 93 (1967) 185. V.N. Bagaev, V.I. Okulov, E.A. Pamyatnikh, Pis’ma Zh. Eksp. Teor. Phys. 27 (1978) 156 [JETP Letters 27 (1978) 1443. V.G. Peschansky, G. Espeho, D. Tesgera Bedassa, Fiz Nizk. Temp. 21 (1995) 971 [Low Temp. Phys. 21 (1995) 7481. 0. Galbova, G. Ivanovski, O.V. Kirichenko, V.G. Peschansky, Fiz. Nizk. Temp. 22 (1996) 425 [Low Temp. Phys. 22 (1996) 3311.
ELSBVIER
Reports
REPORTS
288 (1997) 3055324
Kinetic phenomena in layered conductors placed in a magnetic field Y.G. Peschansky B.I. Verkin Institute for Low Temperature Physics and Engineering, Ukrainian National Academy 47 Lenin Avenue, 310164 Kharkov, Ukraine
ofSciences,
Abstract
A theoretical study on galvanomagnetic, high-frequency and magnetoacoustic phenomena in layered conductors with quasi-2D electron energy spectrum of an arbitrary form is presented. It is shown that the specific character of the quasi-2D energy spectrum of charge carriers causes a series of peculiar effects. It is the orientation effect - the oscillation dependence of kinetic characteristics on the angle between a strong magnetic field and a normal to the layers, specific resonance effects and acoustic transparency of the layered conductors. In a wide range of magnetic fields Kapitza’s law (the linear growth of the magnetoresistance with increasing value of strong magnetic field) is valid. PACS:
72.15.Gd; 73.61.At; 74.70.Kn; 75.70.Cn
Keywords; Layered conductor; electromagnetic field.
Magnetoresistance;
Acoustic
transparency;
Orientation
effect; Penetration
depth of the
1. Introduction A considerable success in the construction of the theory of condensed media is associated with the utilization of the concept of quasiparticles - elementary excitations above the basic state. The electron theory of metals developed by I.M. Lifshitz with the most general assumptions concerning charge carriers’ dispersion law [l] has stimulated an expensive complex of experimental investigations of thermodynamics and kinetic characteristics of metals placed in an external magnetic field. In a strong magnetic field these characteristics, being calculated under the assumption that the dependence of the conduction electron energy E on the momentump is known a priori, turn out to be very sensitive to the form of the electron energy spectrum. As a result the inverse problem formulated by I.M. Lifshitz can be solved by experimental study of the magnetic susceptibility and kinetic effects in metals in a strong magnetic field when the charge carriers rotation frequency C2 exceeds significantly their collision frequency l/r. The problem is to restore the Fermi surface E(P) = EF - the principal characteristic of the electron energy 0370-1573/97/$32.00 0 1997 PII SO370-1573(97)00030-6
Elsevier Science B.V. All rights reserved
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V.G. Peschansky / Physics Reports 288 (1997) 305-324
spectrum. In the momentum space the surface on which the energy is equal to the Fermi energy + turns out to have a complicated form and is open for almost all metals. Galvanomagnetic effects being the most sensitive to the Fermi surface topology have become the basis for constructing the reliable spectroscopic method for studying the topology structure of the electron energy spectrum [2,3]. The well-grounded concept concerning conduction electrons in metals is quite applicable to describe electron properties of a wider category of conductors in which the life time of excitations carrying a charge is large. This idea of I.M. Lifshitz proves to be fruitful for investigations of low-dimensional organic conductors. This is because “disorder” is not a common characteristic for organic conductors and a number of impurities and crystal defects is negligible. In such conductors, at low temperature, free path time of the charge carriers which realize the Fermi branch of the energy spectrum is sufficiently large and the condition fir 4 1 can be realized easily. A considerable part of organic conductors represent layered structures with a sharp anisotropy of electrical conductivity. The conductivity along a normal to the layers n is extremely small and much less than the conductivity along the layers. The latter, as a rule, is of the metal type and because of that the organic conductors are often named “organic metals” or synthetic metals. There are grounds to make use of the quasiparticle concept developed by I.M. Lifshitz for studying transport phenomena in such conductors. The sharp anisotropy of the conductivity of layered conductors is apparently connected with the sharp anisotropy of the velocity of charge carriers on the Fermi surface, i.e. their energy s(P) = C s,(~.+,) cos(anpJh) is weakly dependent on the momentum projection pZ =pn. The coefficients of cosines are assumed to decrease essentially with n, so that Ai = +4o < AI);
A,+1 4 A,;
(2)
where A, is the maximum value of the function E,(P,,P,,) on the Fermi surface, a is the separation between the layers and h the Planck constant. Specific character of the quasi-two-dimensional electron energy spectrum causes a series of the peculiar effects which are absent in the case of ordinary metals. It is the orientation effect - the oscillation dependence of kinetic characteristics on the angle between a strong magnetic field and a normal to the layers, specific resonance effects and acoustic transparency of the layered conductors. We shall consider theoretically galvanomagnetic phenomena and propagation of electromagnetic and acoustic waves through the layered conductors at an arbitrary form of a quasi-twodimensional charge carriers energy spectrum.
2. Galvanomagnetic
phenomena in layered conductors
Experimental observation in a magnetic field about lo-20 tesla of Shubnikov-de Haas oscillations of the magnetoresistance of such layered conductors as the tetrathiafulvalen salts, halogens of tetraseleiumtetracen [4411], indicates that the charge carriers’ mean free path I = m in them is sufficiently large, at least it is about lop3 cm. This gives grounds to suppose that the range of
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strong magnetic fields where galvanomagnetic characteristics are sensitive to the Fermi surface topology, can be reached easily. Below we analyse the dependence of magnetoresistance of the layered conductors on the magnitude of the magnetic field H = (0, Hcos 9, H sin 9) and its orientation with respect to the layers in the quasiclassic approximation, when the corrections to the magnetoresistance connected with the quantization of the charge carriers energy in a magnetic field are small. In order to find out the connection between the current density euif(p) 2d3 p(2xh)- 3 = oij(H)Ej;
ji =
(3)
s
and for the electric field E it is necessary to solve the kinetic equation for the charge carriers distribution function f(P) W + e(vxH)lc)V@
= JJL~{f).
(4)
However, at a small electric field the deviation of the distribution function f(P) =&(E) 11/(P)%&)/aE from the equilibrium Fermi functionfo(e) is small and the kinetic equation (4) can be linearized in a small perturbation of the conduction electrons system. In this approximation the collision integral WColrepresents a linear integral operator acting on $(p), which is the function to be found. At low temperatures when conduction electrons are scattered mainly by impurities and crystal defects we may take the collision integral to be the operator of multiplication by the collision frequency l/z of the unequilibrium correction to the Fermi functionfo(E). So the solution of the kinetic equation is the proper function of the integral operator of collisions. In this approximation the kinetic equation takes the simple enough form h+bjC%,+ $17 = eEv;
(5)
and its solution I,!/= eEi$i = eEi
t dt’ Vi(t’) exp (t’ - t)/z s -‘X
allows to determine the components
(6)
of the electrical conductivity
tensor
where e is the electron charge, tH is the time of electron motion in a magnetic field with the period T = 27c/!CJ according to the equations dpx/dt = (eH/c) { uycos 8 - v, sin 9};
ap,/at = -
(eH/c) u,cos 9;
ap,/at = (eH/c)v, sin 9; (8)
c is the velocity of light, angle brackets denote the and multiplication by 2eH/c (2~/2)~, subscript “H” of t discussion. It is easily seen that the velocity of motion of conduction the momentum projection pH = pZcos 9 + pp sin 9 onto the
integration over the Fermi surface is omitted here and in the following electrons at 11< 1 depends weakly on H-direction, and closed electron orbits
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in the momentum space are almost indistinguishable at different pH. It follows from this that the expansion in a power series with respect to the parameter of quasi-two-dimensionality y1of the conductivity tensor components begins from the quadratic terms if even one of the subscripts at Oij coincides with z [12]. It is easy to see that there are no terms proportional to y in the components giz = -
f n=l
x
7 2e3 Han/c(2n;h)3 h dp, dw(cp,l i i0
exp{(t’ - t)/r} sin {anp,/hcos
8 -
t
s ~ cx
dt'@,
tan 9/h).
anp,(t',p,)
PHI
(9)
The functions E, with n 2 1 is of the order of either ye or higher powers of y. In the linear approximation there is no need to account for the weak p,-dependence of the electron velocity, momentum projection pY and the functions E,. When integrating with respect to pH of the last factor, each of the terms in the sum in the formula (9) vanishes. The asymptotic behaviour of the components Ozj is analogous because in the expression for the velocity Uj(t,PH) = Vj(t) + Atij(t,PH) the corrections which depend on pH are proportional to q, and the velocity projection U, averaged over &, at fixed t is equal to zero in the linear approximation in q. An experimental observation of only two frequencies of the Shubnikov-de Haas oscillations of the magnetoresistance of the layered conductors [4-71 which corresponds to the minimum and maximum Fermi surface cross-sections, indicates that one group of charge carriers in such conductors prevails. The small difference between the frequencies is the result of the small corrugation of the Fermi surface. At least there is no reason to assume the compensation of the electrons and holes volumes in the presence of the several cavities of the Fermi surface to be permissible. When considering the galvanomagnetic phenomena, it is sufficient to take the Fermi surface in the form of a weakly corrugated cylinder with the open direction along the axis pz. The resistance of a conductor along the layers at an arbitrary orientation of a magnetic field is of the same order of magnitude as the resistance of a noncompensated metal, i.e. it differs unessentially from the resistance in the absence of a magnetic field which equals l/o0 q2. The difference from metals shows up in the larger amplitude of the Shubnikov-de Haas oscillations of the magnetoresistance of the layered conductor because the number of charge carriers which form the oscillatory effect increases proportionally to q- 112with decreasing v].At q -%hOI+ almost all of the charge carriers on the Fermi surface contribute to the oscillatory amplitude because all Fermi surface cross-sections of the plane PH are indistinguishable at yl = 0. On the contrary, the resistance of the layered conductor along the “hard” direction, pz,, i.e. along a normal to the layers, is sensitive to the magnetic field orientation. The asymptotic expression for pz,, as a rule, equals to l/o0 and only for some orientations of a magnetic field, at which crZZ< o,&, is determined by a greater number of the conductivity tensor components and increases sharply. Just in this the orientation effect lies. The asymptotic expression for cZZ(q,y) 0
--co
s T
dtl exph/d
dtdt)dt
0
+ tl) , i
(10)
V.G. Peschansb
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/ Physics Reports 288 (1997) 305-324
at yeQ 1 and y = l/L?z < 1 can be analysed easily at the most general assumptions about the function c, which satisfy the condition (2). Restricting our attention to the terms proportional to y2 we integrate over pH easily. As a result we have t
dt’ E,(t) c,(t’) (~n/h)~exp(t - t')/z -72
x [e” H cos S/iia~(27rh)~]cos ((an/h) (p,(t) - p,(t’)) tan $1,
(11)
where all functions under the integral sign depend on t and t’ only. For y < 1 the component electrical conductivity tensor ozz takes the following form fJ
zz
=
ae3zTHcos$/4n2h4cC
n
n21,f + y200{y2~1(9)
+ y2cp2($)},
of the
(12)
where
s I-
Z,(S) =
T-l
dt e,(t) cos {anpy(t) tan 9/h).
(13)
0
The functions qi are of the order of unity and account for them is necessary at the values 9 = 9, at which I1 (9) vanishes. This is the case when the asymptotic expression for the magnetoresistance depends essentially on the velocity of decreasing with n of the functions a,. For example, if I, is proportional to 11”the resistance along a normal to the layers at 9 = SC, instead of saturation in a strong magnetic field, increases proportionally to H2 in the range where y < y < 1 and the saturation of the resistance takes place in a more strong magnetic field at y 5 ye. For tan 9 $ 1 the expression under the integral sign in the formula (13) oscillates rapidly and I,($) can be derived easily by means of the stationary phase method. If there are only two points of the stationary phase where 0, vanishes in an electron orbit the asymptotic expression for Z,(9) takes the form 1,(S)
= 2eJO) (2xhc/aneHv!JO)
sin $1 ‘I2 cos(unD,
tan $/2h - n/4),
(14)
where D, is the Fermi surface diameter along the axis pY, and Pmin = p,(O), v:(O) > 0. Zeros of the function I1 (9) as it follows from the formula (14) repeat periodically with the period A(tan 9) = 2nh/aD,. Additional possibilities for studying the anisotropy of the Fermi surface diameters by the galvanomagnetic measurements are related to the presence of electron orbits strongly along the axis pz which intersect a large number of cells in the momentum space. The motion along such orbits grows as 9 approaches 7c/2 and can exceed z at any value of a field. Therefore, it should be kept in mind that the formulas given above are valid at not values of tan 9. The condition T < z can be satisfied only for electrons on the orbits which the points of self-crossing, for the period diverges as logarithm when approaching tan 9 2 l/q the self-crossing electron orbits appear. Analyse their role for 8 = x/2. The
(15) means of stretched period of magnetic too great have not them. At period of
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electron motion takes the form
along the open cross-sections
s
of the Fermi surface by the plane PH = PY = const.
2nh;a
T(P,) =
dp,
c/eHv,.
(16)
0
The velocity of the charge motion For example av,(t)pt
=
along the open orbit in this case at r] < 1 depends weakly on t.
(d2~~ap:)ap,~at+ (a2E~ap,dp,)ap,~at
= ww) (~,a~,lap, - v,av,iai4
- Y.
(17)
The period of motion on the orbits distant from the self-crossing orbits, is proportional to l/v,(O) within the linear approximation in q (t is supposed to be counted out from the cross section of the Fermi surface pz = 0). However, while approaching the self-crossing orbit pY = pC the velocity along the axis x decreases and account for the small corrections in the parameter v becomes necessary. In the point of self-crossing of the electron orbit pC = (O,p,,O) the velocity v,(O) vanishes and an electron stays near this point for a considerable time. Small value of U, on the electron orbit on which py is closed to pC, corresponds to the weak p,-dependence of E. In this case, we can make use of the expansion of energy in a power series in px while calculating the period of electron motion. Omitting the high-order modes in the formula (1) we obtain a(P) = so(0, P,) + PSR
+ a1(0, P,) cos @PZlh).
Making use of the relation (18) it is easy to calculate orbits close to the self-crossing trajectories
(18) the period
s
of electron
motion
along the
?I
T(p,)
= Y]~“~SZ~’
da(c2 + sin’ a)- 1’2,
(19)
0
where Q. is the frequency to the layers, and
of rotation
of an electron
in a magnetic
t2 = {a - &o(O,P,) - ~l(O>P,))/2~l(O>P,).
field applied parallel to a normal (20)
When approaching the self-crossing orbit, < becomes however small and the integral in the expression (19) diverges proportionally to ln(l/[). In distinction from ordinary metals in which the period of motion of charge carriers is larger or comparable with the free path time only in the small range (of the order of exp{ - sZoz>) of the Fermi surface cross-sections near the self-crossing section, in the quasi-two-dimensional conductor the condition T 2 z is satisfied in a more wide range of electron orbits where t is less or of the order of unity. in ozz. At y”’ 5 y. = 1/C20z the small part of the charge carriers makes the major contribution The velocity along the normal of these electrons can be written in the form v, = - ~~(0, p,) a/h sin C2i t, where fil = aeHv,(O)/hc h/aml.
= C20v,/vF and vr is the characteristic
(21)
Fermi velocity being of the order of
V.G. Peschansky i Physics Reports 288 (1997) 305-324
As it is easy to calculate,
the conductivity
along a normal
311
to the layers has the form
The contribution to crZZdue to electrons on the closed orbits of the Fermi surface is essentially smaller than (22). So the small range of the open orbits near the self-crossing orbit makes the major contribution in the electrical conductivity of a specimen, and the resistance of a conductor along a normal to the layers in the given above range of a magnetic field increases proportionally to H. Further increase of a magnetic field leads to narrowing of the range of electron orbits where the period of motion of an electron is larger than its free path time. As a result the contribution to crZZfrom the electrons near the self-crossing orbits is proportional to y; 2 as well as the contribution of electrons for which T Q r, and at y. < y112 the resistance pZZ increases proportionally to H 2. Thus in layered conductors with the quasi-two-dimensional electron energy spectrum there is a wide range of magnetic fields where the resistance along a normal to the layers increases proportionally to the magnetic field.
3. High-frequency
phenomena in layered conductors
Propagation of electromagnetic waves in layered conductors with a quasi-two-dimensional electron energy spectrum depends essentially on the polarization of the incident wave. A linear polarized wave with the electric field E,(r) along the normal to the layers decays at large distances than a wave with the electric field along the layers. This permits utilizing layered conductors as filters letting the wave through only with a certain polarization. Under the normal skin-effect conditions, when the mean free path 1 of the charge carriers is less than skin depth, the penetration depth 6,, of the electric field E,(r) is l/q times the penetration depth 6, of the electric field along the layers E,(r). When the mean free path of a conduction electron is larger than 61, the relation between 61 and 6,, is 6, =
y2136 II’
(23)
In a magnetic field the wave penetration depth and relation between 6, and 6,, depends essentially on the value and orientations of the strong magnetic field (Gr 9 1). Let us consider propagation of electromagnetic waves along the layers within the x > 0 halfspace in a magnetic field H = (0, H sin 8, H cos 9) parallel to the sample surface x, = 0. The electric field E(x) is to be found using the Maxwell equation curl curlE
- 02E/c2 = 47ciwjlc’;
div E = 4rcp’,
where p’ is the uncompensated charge density. Connection between current density and electric field can be found from the Boltzman equation for the charge carriers distribution function
(24) kinetic
(25)
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We confine our consideration to a linear approximation system. In this case, the kinetic equation is of the form
a+/at, + v,a$/ax + tj(i/~ -
in the weak perturbation
of the electron
iw) = evE(x),
(26)
and current density j is a linear integral operator acting on the function E(x). When solving the Maxwell equation, the wave can be reasonably treated as monochromatic with frequency w. The linearized kinetic equation (26) should be supplemented with the boundary condition allowing for the scattering of charge carriers by the sample surface
where the sample surface specularity W(p,p + ) through the expression &_)
= 1-
d3pW(p,p+)
parameter
s(J_)
is related
to the scattering
(1 - @ [G.(P)]),
indicatrix
(28)
s
where O(c) is the step-function, the electron momentump_ andp. (incident and scattered by the boundary at point x, = 0) are interrelated by the specular reflection condition, which conserves the energy and the momentum projection on the yz plane. The integral term in the boundary condition (27) ensures no current through the sample surface. This functional of the scattering indicatrix should be found with the aid of a rather intricate integral equation [13]. However, in the range of high frequencies co the solution of the kinetic equation depends weakly on this functional and without account for it has the following form at x < (x(tff) - x(0)): $(tfi,hf,
4 =
si
Cl{ dt4t,~FM(4t~~H)
+ 4(hAU
- LH))
- x(i~d)ev(v(t
- 4(khdexp{Ww
-
U)lY
?T-L
X
J i
dtev(t,pH)E(x(t,pH) - x(ApH))exp{v(t-
where v = - ice + l/z, and A is the root of the equation x(t,PH)
- XhpH)
nearest
to
tH
+ 2%- T)},
(29)
tH:
(30)
= x.
For the electrons that do not collide with the sample surface, i.e. for x > {x(tH,pH) - X,in) one should put i, = - CC. Following Reuter and Sondheimer [14], we continue evenly the electric field onto the region of negative x and apply the Fourier transformation to Eq. (24). As a result, the relationship between the Fourier transforms of the electric field and the current density
s cc
E(k) = 2
dx E(x) cos kx,
0
j(x) = 2
Ccdxj(x) cos kx, s0
(31)
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Reports 288 (1997) 305-324
313
take the following form {k2 - Co2/c2}E,(k)
- 4Soj,(k)/c2
= - 2EL(O), CI= (y, z).
(32)
Apart from Eq. (32) the relation between j(k) and E(k) can be found by solving the kinetic equation (26), viz., ji(k) = oij(k)Ej(k)
+
s
where
dk’ Qij(k, k’) Ej(k’),
(33)
t ss s T
aij(k) = 2e3 H/c(~Tc~)~ dp,
dt’ uj(r’,PH)
dtui(t,PH)
0
x
exp{v(t’ - t)}
cos
-cc
k{x(t’,PH)
- x(t,PH)} G (e2ViRUj).
(34)
The integral operator kernel Qij depends essentially on the state of the sample surface. In a strong enough magnetic field parallel to the surface of a conductor, when the trajectory diameter 2~ of electrons along the x axis is much smaller than the skin-layer depth and the charge carriers mean free path 1,the relation between current density and inhomogeneous electric field can be to a good accuracy treated as local and independent on the sample surface condition since the current due to electrons colliding with the sample surface is negligible. In this case the skin depth can be accurately enough determined from the roots of the dispersion equation det{&a - 5G&)/c2}
= 0,
a,B =
(35)
(Y,z),
where t = 4rcio/(k2c2 - 02) and &&) = o,p(k) - o,,(k) o,#lMk).
(36)
With CM5 1 the decay length d1 pertaining to the electric field E,(x) coincides in order of magnitude with & = (oz)-~‘~ c/o0 for an arbitrary relation between 6,, and 1 and for any orientation of the magnetic field within the yz plane, since r?,,(k), to within numerical factors of order unity, coincides with go which is the electrical conductivity along the y axis in homogeneous electric field at H = 0. Here o. is the frequency of plasma oscillations of the electron gas. The electric field decay length 61, along the normal to layers depends essentially on the angle 9 between the vectors IZand H. This stems from the fact that at certain values of 9 there is an abrupt decrease of the expression for ozz(k), when kr < 1 and G?r9 1. The asymptotic expression for o,,(k) in a very strong magnetic field takes the following form: o,,(k) = 2
n21i(9) ue3 H cos Ip/cl/(2~h)~
II=0
+
nor2
{Y~~‘PIW
+
(W2
4n2(9)
+
~~v3(9)),
(37)
where y = v/L?; 1,(g) is given by the expression (13), the functions vi(S) are of the order of unity.
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At Svalues essentially differing from lp, (we recall that Ii (SC) = 0) there exists a universal relation between 6,, and dL, if q > (co/c~)~‘~, namely, to within a numerical factor of order unity
When the Fermi surface corrugation depth 61, = (1 + co2r2/c2)-1’2
is very small, namely,
6; o/c
q < min {(w/ao)1iZ,60/r},
the skin
(39)
increases with increasing magnetic field to reach the limiting value 6; w/c. The penetration depth for the electric field E,(x) grows substantially for 9 = gC, i.e., in the angular dependence of the impedance there is a series of narrow spikes at 9 = SC,, which in sufficiently pure conductors (1~~ > 6,) diminish with increasing magnetic field. On the other hand, they grow proportional l&,/ry, if ly2 < & and ly -=zr < do/y. At not too high frequencies, when the displacement current is small compared to conduction current across the layers, skin depth ~5)~ can be represented by the following interpolation formula (40) In the case of essentially depth 6,1 has the form
low conductivity
611= { 1 + (r/l~)~ + (rco/cq)“}
across the layers, when o > 00y2(q2 + r2/12), the skin
- lj2 &/y2,
(41)
and in a strong magnetic field, when r < (12y2 + 6i/q 2 ) I” , the electric field decay length across the layers is again equal to &/q2. The narrow spikes of the high frequency electric field E, with tan 9 $ 1 periodically appear as a function of the angle 9 with the period (15). Under the extremely anomalous skin effect conditions, when the skin depth is the least parameter of dimension length in the problem, the impedance and transparency of thin plates are sensitive to the state of the surface of a sample. If the surface is rough and it reflects charge carriers diffusely a universal relationship (23) between d1 and 6,, applies in a wide range of magnetic fields. When a specularity reflecting surface is parallel to a magnetic field, the relationship (23) is then valid only in the range of a weak magnetic field (r > l), whereas in a strong magnetic field (r < 1) we have 61 = 6,,qY
(42)
The relationship (42) is valid also in the case of near-specular reflection, when the effective width of charge carriers scattering indicatrix w < r312/18 f,‘2. If w + r3i2/ld:12 and 6, 4 r 4 1, then Eq. (23) applies. In the intermediate case when r312/16i’2 $ w 6 r3’2/1d:12, only 6, depends on w: 6,, = r 1’3(&/Y)2’3,
d1 = (w~)~/~S:/~ rp’j5.
(43)
It therefore follows that measurements of the impedance as a function of the magnetic field can be used to determine the state of the sample and the degree of anisotropy of the Fermi surface.
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315
When the skin layer depth 6 is considerably less than the diameter of an electron orbit in a magnetic field, an electromagnetic field penetrates in a conductor also in the form of narrow spikes, separated by distances that are multiples of the extremal diameter of the electron orbit (15). The weak curvature of the Fermi surface along the pZ axis favours participation of a large number of charge carriers in these spikes. If ye< 6/r, then almost all the conduction electrons with the energy E = FF participate in the penetration of an electromagnetic field deeper into a sample and intensities of such field spikes are of the same order of magnitude, with the exception of the last spike closest to the X, = 0 boundary of a thin plate (d 4 1), opposite to the skin layer, because on approach toward this boundary to a distance A (where 6 6 A 4 r), a small factor (A/r)“6(&,r)2i3 { 1 + wl/r}l13 appears due to the screening of the field in a spike by the current of electrons slipping along the boundary [16,17]. Investigation of the surface impedance in a magnetic field will allow determination in finer detail of the electron energy spectrum in layered conductors and the state of the sample surface.
4. Acoustic transparency of layered conductors When acoustic waves propagate in the layered conductor placed in a magnetic field, the anomalous transparency should be expected as well. Being very sensitive to the form of electron energy spectrum, the magnetoacoustic effects [l&21] have been used successfully for restoration of the Fermi surface, and in the low-dimensional conductors they are worthy of the special examination. In conducting crystals apart from the sound waves attenuation related to the interaction between thermic phonons and coherent phonons with the frequency co. there are many mechanisms of electron absorption of acoustic waves. The most essential of them is the so-called deformation mechanism connected with the charge carriers energy renormalization in the deformed crystal: KE=
AijUij.
(44)
In a magnetic field the induction mechanism - Joule losses - connected with electromagnetic fields generated by sound waves compete with the deformation mechanism. These fields should be derived from the Maxwell equations (24), and connection of the current density with the deformation tensor Uij and electric field E = E + (ti x H)/c + mii/e
(45)
can be found with the aid of the solution of the Boltzman kinetic equation. The field g is determined in the concominant system of axes which moves with the velocity ti. The last term in the expression (45) is connected with the Stuart-Toulmen effect. Together with the Maxwell equations and Boltzman kinetic equation it is necessary to consider the dynamic equation of the elasticity theory for the ionic displacement U. The equation contains the force which represents the influence on the crystal lattice of the electron system excited by the acoustic wave. In a weakly deformed crystal the complete set of equations for this problem is suggested by Silin [22] for isotropic metals and generalized by Kontorovich [23] to the case of an
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V.G. Peschansky / Physics Reports 288 (1997) 305-324
arbitrary dispersion law of charge carriers. A complete set of nonlinear equations for this problem, valid for any wave intensity, was derived by Andreev and Pushkarov [24]. In the approximation that is linear in the powers of the deformation tensor the change of the charge carrier dispersion law (44) can be described with the aid of the deformation potential 3Lijwhose components depend on the quasimomentump only and coincide in order of magnitude with the characteristic energy of the electron system, viz., the Fermi energy CF. Confining ourselves only to the linear approximation in a weak perturbation of conduction electrons under the action of deformation of the crystal in kinetic equation, we obtain for $ the following expression: $ = I?
{Aij(p)
Uij
+
eEv>,
(46)
where I? is the resolvent of the Eq. (27) and /lij(p)
=
&j(p)
-
The condition for existence of nontrivial solution of the complete set of the linearized equations for the problem represents the dispersion relation between the wavevector k and the frequency tr). The imaginary part of the wavevector determines the decrements of the acoustic and sound waves and the real part accounts for the renormalization of their velocities related to the interaction of the waves with conduction electrons. However, the sound attenuation rate can be also determined with the aid of the dissipation function Q which is proportional to the alteration with time of the entropy of a conductor [25]. Taking into account the electron absorption of acoustic waves only, we have for the dissipation function (47) and for the sound damping decrement
r = (I II/12/Pi2 ST),
(48)
where p is the density of the crystal, s the sound velocity, the collision integral is taken in the z-approximation. Using the Fourier representation for the Maxwell equations we obtain the connection between the electric field and the displacement of ions: (49) where /Lij= 6ij - kikj/k2, 6ij is the Kronecker symbol I?, and Ep are the projections of the electric field vector on the plane orthogonal to the sound wavevector. The electric field along k can be found from the condition of absence of the electric current along the sound wavevector, sinj is the anti-symmetrical tensor (s123 = 1). The components of the tensor oij(k) and aij(k) which connect the Fourier transformations of the current density, electric field and ionic displacement ji(k) = oij(k) Ej(k) + aij(k) kouj(k),
V.G. Peschansky 1 Physics Reports 288 (1997) 305-324
317
are of the following form: CJij(k)
=
(e2ViRUj);
aij(k)
=
(euiRAjn)
ujk”/k.
(51)
The components a,j and GaBconnect j, with Uj and Ep taking into account the absence of the current along the wavevector. Let us consider an acoustic wave propagating in the layers-plane in the x-direction orthogonal to a magnetic field H = (0, sin 9, cos 9). In this case when the parameter yeis small the solution of Eq. (49) takes the form: E,(k) = (1 - ~6yy(k)}-1 [icou,Hcos9/ck2
+ lkoiiyj(k)uj],
i?‘,(k) = - icou, H sin S/ck2 + ma’u,/e,
(52)
(53)
where ii,j(k) = aij(k) - a,j(k) ai,(k)/o,,(k). Expression (53) is valid when y2 < /3 = (oc/coO s)“/coz. In the opposite case, in order that E, be determined it is necessary to account for the dependence on q terms in the expansion of j, in a power series of the small parameter q. Nevertheless, at however small y there is no need to account for EZ because 0, E, is proportional to q, and it is sufficient to take into consideration the electric field E, only when calculating the asymptotic expression for the dissipation function at q = 0. In ordinary metals Joule losses are essential in the range of strong enough magnetic fields when the radius of curvature of the electron trajectory is much less than not only its mean free path but also the sound wavelength, i.e., kr 4 1. If the charge carriers trajectories are bent not too strongly, so that 1 < kr < kl,
(54)
the absorption of the sound wave energy in a metal is determined mainly by the deformation mechanism. In low-dimensional conductors the role of electromagnetic fields generated by a sound wave turns out to be essential in a more wide range of magnetic fields, including a magnetic field which satisfies the condition (54). In this case acoustoelectronic coefficients aij and Oij oscillate with an inverse value of a magnetic field and their asymptotic behaviour at 1 4 kr < l/q and P < I takes the following form: 6Jk)
= (G/kD)(l - sin kD);
a”yj(k) = - i(G/lj,/evkD)co~ kD,
(55)
where D = cD,/eH cos 9 and D, is the diameter of the Fermi surface along the p,-axis, 21and /ij, are the electron velocity and value of ijx(p) at the reference point along the p,-axis, G = 4vD,e2z/ac(2nh)2. At 1 < kD 4 Z/ythe asymptotic expression for the dissipation function of the sound wave with the
longitudinal
Q=
polarization
takes the form:
eHz cos S{g?(l + sin kD) + g$(l - sin kD) - 29, g2 cos kD) n2h2 akvc{l + ~~~,,,~‘}
where g1 = Aj,UjUk and g2 = VCOU,~Hcos $1~.
(56)
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/Physics
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It is easily seen that the asymptotic behaviour of the acoustoelecronic coefficients gYv and a”yjalters essentially at kD = 2n(n + i), which leads to the sharp increase of the dissipation function. However, out of the resonance, i.e. when cos kD differs from zero essentially and there is no need to take into account small corrections in the formulas (55) for the acoustoelectrical coefficients, the denominator in the expression (56) for the dissipation function increases with a magnetic field proportionally to H2 and the sound damping decrement has the form r E (co/v)r/l.
(57)
In this case the decrement decreases with increasing of the magnetic field magnitude and the mean free path of charge carriers. The attenuation length for a longitudinal wave is very large, when kD is close to 2~c(n- b), because g1 E g2 kD is much larger than g2. As a result, in addition to resonance we can expect the emergence of acoustic transparency when the path between two points of a stationary phase changes by half of the wavelength. As a result, between resonant values of the sound decrement which repeat with the period A (l/H) = 2ne cos $/kcD,
(58)
the anomalous acoustic transparency should be observed with the same period. In sufficiently pure conductors at low temperatures the charge carriers mean free path can be so large that the condition kln $ 1 is satisfied, there is the range of a magnetic field where l/q < kr + kl, and the oscillations of the acoustoelectric coefficients are formed by not all of charge carriers on the Fermi surface but by the small part of them. These are the charge carriers near the Fermi surface cross-section which has the extreme diameter Dixt. In this case the magnetoacoustic resonance at kl_H is absent, and the sound decrement T(H) = (1 + (kry)-‘12 sin(kD, - r-c/4)}ozjr
(59)
is of the same order of magnitude as in an ordinary metal. Unimportant numerical factors in the formulas (57) and (59) are omitted and Do = cD,““‘/eH cos 9. The magnetoacoustic resonance is possible at k_LH only if kry < 1. In the expressions for g,,v it is necessary to take into account the small corrections in the parameters r/l and l/kr near the resonant values of a magnetic field. It is not difficult to obtain the following formula for T(H) in the resonance:
At 1 < kr2 the absorption of the sound wave energy by conduction electrons increases proportionally to H with increasing magnetic field. The absorption of transverse waves in a layered conductor depends essentially on the concrete form of the nondiagonal components of the deformation potential tensor /lij(p). In the cases when in the range of the effective interaction between charge carriers and the sound wave where kv = CO,& and /i,, are much less than A,,, the order of magnitude of the sound damping decrement changes essentially. The periodic alternation of the transparency and the resonant absorption of the shear waves polarized in the layer-plane is of the same character as in the case of longitudinal waves.
V.G. Peschansb
1 Physics Reports 288 (1997) 305-324
319
Acoustic waves polarized along a normal to the layers attenuate at considerably larger distances if the component /1,, is small and vanishes at q + 0. In this case the solution of the Maxwell equations takes the form E
=
Y
kwii,,
+
mw2cYy,/e
1 - Gyy
E, =
4%
a",,ko(
+ mco”/e
1 - &?,,
u.
It is easy to see that the components of the matrix aij as well as oij do not contain the linear in y terms if at least one of the subscripts i or j coincides with z. When calculating the dissipation function, there is no need to retain the electric field E, which is proportional to y2 because ev,E, contains the term proportional to the first power in r. When 1 < kr <
Q=
co3m2eH cos 9 acsv(27ch)2
iAi ammo
/1,, +
hk(l - 55:,,)
cos (aD, tan 9/h) 2 (1 + sin kD).
When calculating Q we restrict ourselves to the first and second terms in the expression (1) for the charge carriers dispersion law. The acoustic transparency takes place when kD equals strictly to 27c(l - 4) and it is necessary to retain the small corrections in “/and l/kr. If sin kD differs essentially from ( - 1) the attenuation of sound waves increases with increasing H as so as in ordinary metals. The attenuation is determined mainly by the first term in the brackets in the formula (62) except for some exotic models of the deformation potential for which /1,, is much less than q&F.The amplitude of the oscillations of r with the period (58) which are caused by the periodic dependence of f7ZZ= y2go/kl { 1 + sin kD cos (uD, tan S/Jr);
(63)
is less than the smooth part of r in v/s times if /1,, r q&F.Besides, the decrement oscillates with 9 with the period (15), tan 9 need not be much more than unity. In the magnetic field conditioned by (54) the angular oscillations take place in the whole range of angles between the magnetic field vector and a normal to the layers. If an electron drifts along the sound wavevector (for instance, the sound wave propagates along the y-axis) the sound decrement reduces in (klq)2 times for r/l < krq < 1. The solution of the kinetic equation in this case taken the form t+T I) =
{exp(vT + &CT) - l}-’
t
s
dt’g(t’)exp(ik(r(t’)
- r(t)},
(64)
where g(t) = /Iji(t)kiuj + ev(t)E”. At 1 Q kly < l/r in the expansion in the powers of VT and kS_T = jt dt b(t) of the factor which is in front of the integral, the terms proportional to k are the most essential. Finally, in the case when the charge carriers drift along k with the velocity Vy= V,tan 9 E qv tan 9 should be replaced VT by kry tan 9 in the expression for the dissipation function. If kry tan 9 9 1, i.e. for the free path time an electron is capable to drift along the sound wavevector at distance which exceeds significantly the sound wavelength, the magnetoacoustic resonance predicted and studied theoretically by Kaner, Privorotsky and the authors of this paper [20] takes place. The resonance occurs at 6T and in contrast to an ordinary metal the amplitude of the resonant oscillations is determined by the parameter krq rather than kr.
KG. Peschansky / Physics Reports 288 (1997) 305-324
320
The formulas given above are valid when cos 9 $ cD,/eHl. If 9 is close to rc/2, i.e. cos 9 is so small that an electron have no time to make a total rotation along the orbit in a magnetic field, then the components of the tensors (T,,,,and crZZare closed to their values in the absence of a magnetic field. This results from the fact that in the quasi-two-dimensional conductor only the projection H, effects on the charge carriers dynamics, and at the q 4 1 the component H, manifests itself only in small corrections in the parameter ye. At 9 = 42 the dependence of the sound damping decrement on the magnetic field magnitude contains only the terms which vanish when r~+ 0, and the magnetoacoustic effects are pronounced in the case of the shear wave with the ionic displacement along a normal to the layers only. In the range of sufficiently strong magnetic field (kr Q 1) the attenuation rate for the wave with such polarization depends essentially on the magnitude of a magnetic field and its orientation with respect to the layers, and in the S-dependence of r sharp peaks and valleys appear. At tan 9 9 1 they are repeated periodically with the period (15). The concrete form of the 9dependence of r is in a great measure analogous to the angular dependence of the electromagnetic impedance for kr < 1. When the acoustic waves propagated along the normal to layers the Maxwell equations have the form {1 - 56:,,(k)} E, - 56,,(k) B, = rfi,j(k) Uj - (u, H, - u, HY) io/c - mo2u,.e; -
t&,,(k) E, + { 1 - 55,,(k)} Ey = [a”yj(k) Uj + u,H, io/c - mo2u,/e.
It is easy to see that the electric field and the components of the matrix gGlado not vanish when q - 0 and, consequently, the induction mechanism of the sound waves attenuation is more essential. The drift of conduction electrons along the z-axis does not take place only for the magnetic field orientation in the layers-plane. If 9 is not equal to 42, at krq Q 1 the displacement of charge carriers along the wavevector for the period of motion in a magnetic field is much less than the sound wavelength and the acoustoelectronic coefficients are of the same order of magnitude as the analogous values for the case of weak spatial dispersion being reduced in klq times if kly p 1. The essential dependence of r on H occurs only in the range of magnetic fields when krq % 1. At tan 9 < leH/cD, an ordinary magnetoacoustic resonance takes place. The resonance is connected with the charge carriers drift along the sound wavevector. At 9 = 42 electrons drift in the plane xy only, i.e., the direction orthogonal to the wavevector. In this case the sound attenuation rate oscillates with l/H which is analogous to the Pippard oscillations [21] in metals T(H) = ycoz/r { 1 - (krr])- “’ /?sin(kcdD,,/eH
Measurements
+ 7c/4)}.
(66)
of the period of these oscillations
d(l/H) = 4ne/kcdD,,,
(67)
enable the corrugation of the Fermi surface to be evaluated. Here d is the difference between the maximum and minimum diameters along the axis pXat pY= 0. The condition kry % 1 is very strict and can be satisfied in the range of a magnetic field where r <<1 only for q 2 &. Therefore, there are no grounds to expect that the clear dependence of r on the magnitude and orientation of a magnetic field can be observed in the layered conductors nowadays.
V.G. Peschansky /Physics
321
Reports 288 (1997) 305-324
Charged elementary excitations in conductors form a Fermi liquid, and their energy spectrum is determined by the distribution function for quasiparticles. As a result, the response of the electron system in solids to an external perturbation depends to a considerable extent on the correlation function describing the electron-electron interaction [27,28]. Usually, the inclusion of the Fermiliquid interaction of charge carriers leads to a renormalization of kinetic coefficients calculated under the assumption that conduction electrons form a Fermi gas. In some cases, however, the Fermi-liquid interaction approach leads to specific effects such as spin waves in nonmagnetic metals [29] and “softening” of metals in a strong magnetic field [30]. We shall show that the inclusion of the Fermi-liquid interaction of charge carriers depends significantly on the shape of the resonance curve and the value of acoustic transparency, but it does not violate the position of peaks of the acoustic wave absorption as a function of the value and orientation of a magnetic field [31,32]. The energy of elementary excitations carrying a charge has the form E = E(P) +
iLij(p)
Uij
+
Y(p,
Y,
t),
(68)
where the last term in this formula takes into account the correlation electronelectron interaction
effects associated with
Y b, r, t) = 2(27ch)- 3 @(p,p’) 6f(p’, Y,t) d3p’,
(69)
s
where 6f=f(p,r, t) -fo{~(P)} is th e nonequilibrium correction to the Fermi distribution function fo{s(P)} for charge carriers in the undeformed conductor. The Landau correlation function @(p,p’) can be expanded into the complete set of orthonormal functions 4”(p): (70) Confining ourselves only to the linear approximation in a weak perturbation of conduction the following equation for the nonequilibrium correction electrons we obtain - $(p,u)exp{ - iwt} 8fo(s)/& to the equilibrium distribution functionfo(s) for charge carriers in the concomitant reference frame moving with the velocity of ions: vrl/ + varl//ar +
a$/at, =
evE - i0 {
Aij
{p}
Uij
+
Y
(p,
r)}.
(71)
Using the Fourier representation y(P, 4 = g dk K(k) 6, (P) cxp {ikr) n=Os
(72)
we obtain the following system of algebraic equations for the Fourier transforms YJk): Y,,(k) (1 + @i ‘> + ia =
-
ikjui(k)(&(p)
(
hh-4~
Aij(p)>
f n=l
+
($n(P)
4A.p)
Y,,,(k) >
aCevg:(k)
+
kjaA,j(p)
u,(k)]),
(73)
322
V.G. Peschansky /Physics
Reports 288 (1997) 305-324
where {ik[v(t’) - r(t)] + v(t’ - t)}.
‘dt’g(t’)exp
Rg =
(74)
s
The value of wr is smaller than unity even in pure conductors at low temperature in a wide acoustic frequency range, and the integral term in formula (73) can be taken into account in the perturbation theory. In the asymptotic approximation in the small parameter OX, the Fourier transform of the kinetic equation solution $(k) assumes the form $(k) =
R{t?Ej(k)Uj
+
kiCO/iji
#j(k)}
@,(l +@“)-i [(4nReEjUj)
- ioR” z
- ikiuj(+,Aji)
+ o($nAji)kiUj]
+n
(75)
n=l
and the acoustoelectric coefficients gij(k) =
(e2viR”jV)- ioe2 $ @,(l + @,))I (ViR4n) (4nR”vj), ?l=l
Uij(k) = (eviR”Aj,)
km/k -
f
@,(I + @n)-’ (evi&J
(76)
((4nAjm)kmlk
n=l
+
im(4,i?Aj,
> k,/k}.
(77)
Using the expressions (76) and (77) we can easily determine the damping decrement for the acoustic wave. For example, we consider the influence of Fermi-liquid interaction on the attenuation rate of sound in the case when k = (k,O,O) and displacement of ions is in the plane xy. For brevity of computations only, we assume that 4r (-p) = 4l (p) and c$~(-p) = - ~$~(p),while the remaining & with YE> 2 are equal to zero. Taking into account that a( -p) = E(P), at ye< krq -@ 1 we obtain for rrY,,the following expression: liquid gYY
=
Ggas yy
(1
-
(78)
L),
where L
=
ioQ
A$:(1
+sinkD)+&4:(1
-sinkD)},
(79)
1
and c#J~,cj2 and v are the values of the functions 4i(t) and the velocity modulus when ku = CO. Naturally, the acoustic transparency of layered conductors with a quasi-two-dimensional electron energy spectrum and the sound attenuation rate depend on the intensity of the Fermi-liquid interaction of charge carriers, but the period of oscillations of r with l/H and the positions of sharp peaks in the angular dependence of r(9) remain unchanged when Fermi-liquid effects are taken into account.
V.G. Peschansky /Physics
Reports 288 (1997) 305-324
323
5. Conclusions The concept of quasiparticles developed by I.M. Lifshitz to describe electronic properties of metals have been used for investigations of kinetic phenomena in layered conductors with a quasi-two-dimensional electron energy spectrum. It is shown that the quasi-two-dimensional nature of the spectrum leads to the specific effects in a magnetic field. The magnetoresistance of layered conductors in a strong magnetic field directed in the plane of the layers can increase proportionally to a magnetic field value. The electromagnetic impedance and the sound attenuation rate depend essentially on the polarization of the incident wave and the angle between a magnetic field and a normal to the layers. Propagation of electromagnetic and acoustic waves in these conductors involves virtually all charge carriers in the transfer of acoustic pulses and electromagnetic field spikes to the bulk of the conductor. The orbits of Fermi electrons in a magnetic field are virtually indistinguishable, which allows to include a large number of conduction electrons in the formation of peculiar oscillatory and resonant effects which are absent in the case of ordinary metals. Investigation of these effects open up the possibilities of studying in detail the dissipative processes in the electron system of layered conductors and the charge carriers’ energy spectrum.
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